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So I have field $F$ (any characteristic), and its prime subfield $K$.

I have three definitions: (i) that $K$ is the subfield of $F$ such that $K$ has no proper subfield; (ii) that $K=\bigcap_i K_i$ where $\left\{K_i\right\}$ is the set of all subfields of $F$; and (iii) $K$ is a field generated by the unity of $F$.

I could reconcile the first two definitions, but could not with the third. How do I show that a prime subfield is in fact generated by a unity?

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By definition, the subfield generated by a set $S$ is the intersection of all subfields containing $S$.

Now, if you take $S=\{ 1\}$, you are intersecting all subfields, since all subfields contain $1$ by definition.

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